Question

If theta is the acute angle between the lines
3x² + 4xy + by^2 = 0 and tan theta = 1/2 . Find b.​

Answer 1

The value of b is (- 55), 1

Formula: Let us consider a pair of straight lines

ax² + 2hxy + by² = 0

If θ be the angle between the two straight lines, represented by the pair, then

tanθ = {2√(h² - ab)}/(a + b)

Solution: The given pair of straight lines is

3x² + 4xy + by² = 0 ..... (1)

Given that, θ is the angle between the straight lines, represented by (1). Then

tanθ = {2√(2² - 3b)}/(3 + b)

= {2√(4 - 3b)}/(3 + b)

Given, tanθ = 1/2

or, {2√(4 - 3b)}/(3 + b) = 1/2

or, 4√(4 - 3b) = 3 + b

or, 16 (4 - 3b) = 9 + 6b + b²

or, 64 - 48b = 9 + 6b + b²

or, b² + 54b - 55 = 0

or, (b + 55) (b - 1) = 0

Either b + 55 = 0 or, b - 1 = 0

Therefore b = - 55, 1